C₇₉×#12061 #34123750

order	20224
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₇₉ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₇₉×#12061

Subgroups

	subgroup	G/H
center	C₂×C₈×C₇₉	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₇₉
Frattini, cosocle	C₄×C₈	C₂³×C₇₉
socle, prime radical	C₂²×C₇₉	C₈×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₇₉ ◁ C₄×C₁₆×C₇₉ ◁ G

200 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 114 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 111
transitive degree	μ_t(G) = 20224

A minimal faithful permutation representation — 3 generators on 111 points:

(1,2,5,9,6,3,10,12,4,7,13,15,14,11,8,16)(17,18)(19,25)(20,28)(21,23)(22,26)(24,30)(27,32)(29,31)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111);(2,7)(3,11)(9,15)(12,16)(17,19,21,27)(18,22,23,31)(20,29,30,26)(24,32,28,25);(1,3,5,12,6,7,10,15,4,11,13,16,14,2,8,9)(18,23)(22,31)(24,28)(25,32)

Conjugacy classes & character table

5056 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 5049);